Abstract In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal., 50, 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme. 

An unconditionally energy stable finite element scheme for a nonlinear fluid–fluid interaction model

Second order unconditional linear energy stable, rotational velocity correction method for unsteady incompressible magneto-hydrodynamic equations

In this paper, we study a fully discrete vector penalty-projection method for the time-dependent incompressible magnetohydrodynamics flows. This fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization and first-order backward Euler for temporal discretization. Moreover, unconditionally energy stable is established, and error estimates for the fully discrete scheme is also derived. Finally, some numerical experiments are provided to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme. 

A vector penalty-projection approach for the time-dependent incompressible magnetohydrodynamics flows

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

A Fully Discrete Decoupled Finite Element Method for the Thermally Coupled Incompressible Magnetohydrodynamic Problem

Second order fully decoupled and unconditionally energy-stable finite element algorithm for the incompressible MHD equations

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Second order fully decoupled and unconditionally energy-stable finite element algorithm for the incompressible MHD equations